^{1, 2}

^{1}

^{1}

^{2}

The global dynamics of discrete competitive model of Lotka-Volterra type with two species is considered. Earlier works have shown that the unique positive equilibrium is globally attractive under the assumption that the intrinsic growth rates of the two competitive species are less than 1+ln 2, and further the unique positive equilibrium is globally asymptotically stable under the stronger condition that the intrinsic growth rates of the two competitive species are less than or equal to 1. We prove that the system can also be globally asymptotically stable when the intrinsic growth rates of the two competitive species are greater than 1 and globally attractive when the intrinsic growth rates of the two competitive species are greater than 1 + ln 2.

In this paper, we further consider the global dynamics of discrete Lotka-Volterra model

The discrete Lotka-Volterra models have wide applications in applied mathematics. They were first established in biomathematical background and then have proved to be a rich source in analysis for the dynamical systems in different research fields such as physics, chemistry, and economy [

Model (

It is well known that for the single-species Logistic model

Our aim of this paper is to obtain some global dynamics of (

The paper is organized as follows. We give some preliminaries in Section

A pair of sequences of real positive numbers

If a solution of (

Assume that

Solving the following scalar equation system:

The equilibria

Denote

For any fixed

(1) If

(2) For any positive sequences

One can refer to [

Next we give some definitions that will be used in this paper.

System (

System (

The solution

Suppose that

The positive equilibrium solution

The following lemma can be found in [

Consider the following difference system:

there exist positive constant

system (

for any positive solution

Then system (

In this section, we will obtain the permanence, global attractivity, and global asymptotical stability of system (

For every positive solution

Note that

Assume that

The proof of this lemma is similar to that of [

Note that

Assume that (

Assume that (

If we denote

Note (

From (

At this point, we claim that

It is easy to verify that the function

We rearrange the two equations of (

Similarly, we have

The proof of Case (iii) is similar to that of Case (ii).

We have

The proof of claim (

Assume that the assumptions of Theorem

From Theorem

Let

Assume that (

From the proof of Theorem

In this section, we give two numerical examples to show the feasibility of the assumptions of the results. The first example also shows that system (

Consider the following case of system (

We see that the conditions of Theorem

The following example shows that system (

Solutions of system (

Consider the following case of system (

It is clear that the conditions of Theorem

Example

Solutions of system (

In this paper, we further discuss the global dynamics of a discrete autonomous competitive model of Lotka-Volterra type. Sufficient conditions are obtained to guarantee the permanence, global attractivity, and global asymptotical stability of the system. These conditions are expressed by the coefficients of the model and can be easily verified. Numerical examples are also given to show the feasibility of the conditions.

Earlier works have shown that the system of this type can be globally attractive when the intrinsic growth rates of the two species are less than

For the global stability of the system, the following condition in Theorem

The authors would like to thank the Editor Professor Yong Zhou and the referees for their valuable comments and suggestions. This work was supported by the Foundation of Jiangsu Polytechnic University (ZMF09020020).